>OK. So here goes changed script for review:>>"For years mathematicians are struggling to prove that they>will always find larger and larger cases of p where p and p+2 >both are primes.>>Someone recently proved that>**>there are as many prime numbers p and q less than 70,000,000>apart as you want>**

Yes, In a sense.

The phrase "as many as you want" can be interpreted as"infinitely many" (provided you always want more and more).

>So, now mathematicians will work on finding what types of >p's this 70 million is negotiable to smaller numbers, >eventually going down to 2."

It's not likely that they'll discover a classification forthe primes which have nearby successor primes.

They simply need to show, for a given integer d >= 2, that there are infinitely prime pairs p,q with p < q <= p + d.

But yes, the goal is to keep reducing the value of d for whichthey can prove the existence of infinitely many such prime pairs,all the way down to d = 2, if possible.